Obections to Cantor's Theory (Wikipedia article)
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From: Han de Bruijn on 29 Jul 2005 04:29 Patrick wrote: > The notion of subset, is distinct from the notion of member. > > {} is a subset of {}. > > {} is not a member of {}. I know that. But I'm changing the rules of the game in accordance with the mantra: A little bit of Physics would be NO Idleness in Mathematics Then {} is a member of {} . > The principal of extentionality says that two sets > are equal iff they contain exactly the same members. > You can't, under ordinary rules, claim that {{}} = {}, > since the LHS has 1 member, and the RHS has none. Under the new rules {{}} = {} . > As far as getting physical goes, there is a simple > physical description of this situation: > > You have the following items: > > (1) a box that contains a box > (2) an empty box > > (1) and (2) don't contain the same things physically, do they? > You wouldn't say they are equal would you? Mathematicians keep saying this, but the box is _not_ a member of the set, it's just the curly brackets that delimit the set. Therefore (1) and (2) _do_ contain the same things physically. > This sort of reasoning, in addition to working well, has > a nice physical interpretation. It doesn't work well and it hasn't a nice physical interpretation. > Well - what do you propose? That people shouldn't be allowed to > do set theory? Different kind of more down-to-earth set theory. But I prefer the magic word instead of dictatorship. Han de Bruijn
From: Han de Bruijn on 29 Jul 2005 04:40 quasi wrote: > Let me try to use the box concept to clear up your (possibly > deliberate) misunderstanding. It _is_ a deliberate "misunderstanding". My excuse is that I intend to play a game that is serious, nevertheless. > View sets as analogous to boxes, and at least for finite sets, the > number of objects inside the box is the number of elements. As I have pointed out elsewhere in this thread, that box argument is invalid, because the box {} is not a member of the set. [ ... rest snipped ... ] Han de Bruijn
From: Robert Kolker on 29 Jul 2005 04:41 Han de Bruijn wrote: > > > Glad that we have such a bright spirit here that it challenges Newton's. Newton got time wrong too, as Einstein showed. Bob Kolker
From: Robert Kolker on 29 Jul 2005 04:44 Han de Bruijn wrote: > > Set theory IS supposed to be an idealization of a physical universe. No it isn't. Set theory is about the idea of collects of things, which need not be physical at all. There is no direct connection between set theory and the physical world. Once again you fail to realize that mathematics as such is not empirical. It does not deal with real physical things at all. Just becase some mathematics can be interpreted and deployed as a tool to be uased to solve physical problems does not mean that mathematics is empirical. Mathematics is about ideas, not physical objects. Bob Kolker
From: Han de Bruijn on 29 Jul 2005 04:48
malbrain(a)yahoo.com wrote: > Since I am a materialist: mathematics is created. Well, maybe it is both discovered as well as created. Karl Marx would have agreed with us, anyway. :-) Han de Bruijn |